Finite element modeling of large deformation response of ...852/fulltext.pdfconcrete (RC) beam was...

FINITE ELEMENT MODELING OF LARGE DEFORMATION

RESPONSE OF REINFORCED CONCRETE BEAMS

A Thesis Presented

by

Andre Werner

To

The Department of Civil and Environmental Engineering

in partial fulfillment of the requirements

for the degree of

Masters of Science

in

Civil Engineering

in the field of

Structural Engineering

Northeastern University

Boston, MA

August 2008

NORTHEASTERN UNIVERSITY

GRADUATE SCHOOL OF ENGINEERING

Thesis Title: Finite Element Modeling of Large Deformation Response of

Reinforced Concrete Beams

Author: Andre Werner

Department: Civil and Environmental Engineering

Approved for Thesis Requirement of the Master of Science Degree

_____________________________________________________ ____________ Thesis Advisor: Mehrdad Sasani, Ph.D., P.E. Date

_____________________________________________________ ____________ Thesis Reader: Dionisio Bernal, P.h.D Date

_____________________________________________________ ____________ Department Chair: Thomas C. Sheahan, Sc.D, P.E. Date

Graduate School Notified of Acceptance

_____________________________________________________ ____________ Director of the Graduate School: Yaman Yener, Ph.D Date

i

ABSTRACT

In a previous study, a 3/8 scaled physical model of a continuous two-span reinforced

concrete (RC) beam was constructed and tested. The RC beam was subjected to a

downward vertical displacement. In this study a detailed three dimensional finite element

model of the RC beam is developed. Unlike a previous analytical study, the contribution

of bond to the response of the beam is considered here by deploying interface elements

between longitudinal reinforcement elements and concrete elements of the finite element

model. Nonlinear material behavior such as concrete cracking and crushing and steel

reinforcement yielding are also modeled. A geometrically nonlinear incremental static

analysis using the computer program DIANA 9.2 is carried out. The model is analyzed

under the applied deformations at the centerline. Analysis results include element forces,

stresses and strains of the concrete and steel reinforcement and bond stresses and slippage

along the beam. Stress-strain relationships of steel reinforcement and beam end rotations

are compared with the corresponding experimental results. Overall, the analytical and

experimental results are in good agreement.

Table of Contents

Abstract i

Table of Contents ii

List of Figures iv

List of Tables viii

Acknowledgements ix

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Spatial Discretization and Element Topology 3

2.1 Cross Section Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 3-D Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Interface between Longitudinal Reinforcement and Concrete . . . . . . 6

2.2.2 Transverse Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Material Constitutive Models 7

3.1 Steel Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1.1 Theoretical Description . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1.2 Material Parameters of Steel Reinforcement . . . . . . . . . . . . . . . 9

3.2 Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 2-D Modified Compression Field Theory . . . . . . . . . . . . . . . . 10

CONTENTS iii

3.2.2 3-D Modified Compression Field Theory . . . . . . . . . . . . . . . . 12

3.2.3 Concrete in Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.4 Concrete Material Parameters . . . . . . . . . . . . . . . . . . . . . . 16

Concrete Compressive Stress-Strain Relationships . . . . . . . . . . . 16

Concrete Tensile Stress-Strain Relationship . . . . . . . . . . . . . . . 19

3.3 Interface between Concrete and Steel Reinforcement . . . . . . . . . . . . . . 21

3.3.1 Mechanics of Interaction between Steel Reinforcement and Concrete . 21

3.3.2 Review of Existing Bond Stress-Slip Relationships . . . . . . . . . . . 23

3.3.3 Parameters influencing Bond Performance . . . . . . . . . . . . . . . . 27

3.3.4 Implemented Bond Stress-Slip Relationships . . . . . . . . . . . . . . 29

4 Analytical Results 35

4.1 Analytical Results at Material Level . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Bond Slip and Section Deformation . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Element End Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 State of Strain at Beam Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Comparison between Analytical and Experimental Results 60

5.1 Steel Reinforcement Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2 Beam End Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Concluding Remarks 63

References 65

List of Figures2.1 View of the RC beam reported in [1] . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Cross section A-A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Cross section B-B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.4 Discretization of the cross section of the RC beam reported in [1] . . . . . . . . 4

2.5 3-D finite element model of the RC beam reported in [1] . . . . . . . . . . . . 5

2.6 Topology of solid element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.7 Topology of interface element and associated nodal displacements . . . . . . . 6

3.1 Uniaxial stress-strain relationships of longitudinal and transverse steel rein-

forcements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 2-D state of stress in a reinforced concrete element [7] . . . . . . . . . . . . . 11

3.3 2-d state of strain in a reinforced element [7] . . . . . . . . . . . . . . . . . . . 11

3.4 Uniaxial compressive stress-strain relationship for concrete cover, side support

and center column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.5 Effectively confined concrete core . . . . . . . . . . . . . . . . . . . . . . . . 19

3.6 Uniaxial compressive stress-strain relationship for concrete core [11] and the

trilinear approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.7 Uniaxial tensile stress-strain relationship of reinforced concrete [10] . . . . . . 20

3.8 Qualitative local bond stress-slip relationships [13] . . . . . . . . . . . . . . . 21

3.9 Transverse cracks originated at the tips of the steel reinforcement ribs and

crushed concrete in front of the ribs of the rebar [12] . . . . . . . . . . . . . . 22

3.10 Radial component of bond stress balanced by tensile ring stresses in the sur-

rounding concrete [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.11 Radial cracks induced by tensile ring stresses and transverse reinforcement ac-

tion [13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.12 Bond stress-slip relationships [12], [15] and [17] . . . . . . . . . . . . . . . . 27

LIST OF FIGURES v

3.13 Defined regions of bond in the finite element model . . . . . . . . . . . . . . . 30

3.14 Pattern of the bond stress-slip relationship in region A and B . . . . . . . . . . 30

3.15 Hydraulic pressure analogy taken from [14] . . . . . . . . . . . . . . . . . . . 31

3.16 Bond stress-slip relationship of interface elements in region A and B as imple-

mented in the finite element model . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 Designation of longitudinal views used for contour plots . . . . . . . . . . . . 35

4.2 Locations of concrete C, steel S and interface I elements . . . . . . . . . . . . 35

4.3 Variation of concrete longitudinal stress σc,zz at cross sectional location C3 at

0.15, 0.3 and 0.48 in of vertical displacement . . . . . . . . . . . . . . . . . . 37



4.5 Longitudinal profile 1 of first principal stress σc,1 [psi] at three vertical dis-

placements: 0.76 (Top); 1.0 (Middle) and 1.2 in (Bottom) . . . . . . . . . . . . 38






0.76, 1.0 and 1.2 in of vertical displacement . . . . . . . . . . . . . . . . . . . 40


0.76, 1.0 and 1.2 in of vertical displacement . . . . . . . . . . . . . . . . . . . 40

4.10 Longitudinal profile 1 of third principal stress σc,3 at 1.2, 1.45, 1.68 and 1.9 in

(plots from top to bottom) of vertical displacement (the face of the side support

is located on the left, the face of the center column is located on the right) . . . 41

4.11 Longitudinal profile 2 of third principal stress σc,3 at 1.2, 1.45, 1.68 and 1.9 in

(plots from top to bottom) of vertical displacement (the face of the side support

is located on the right, the face of the center column is located on the left) . . . 41

4.12 Variation of steel longitudinal strain at cross sectional location S1 at 4.44 and

5.26 in of vertical displacement (horizontal dashed lines represent yield strains) 42

4.13 Variation of steel longitudinal stress at cross sectional location S1 at 4.44 in and

5.26 in of vertical displacement (horizontal dashed lines represent yield stresses) 43

4.14 Variation of steel longitudinal strain at cross sectional location S2 at 4.44 and

5.26 in of vertical displacement (horizontal dashed lines represent yield strains) 43

LIST OF FIGURES vi

4.15 Tensile stress-strain of the first three steel reinforcement elements from the

face of the center column at cross sectional location S2. (Uniaxial stress-strain

relationship of steel reinforcement as implemented in the finite element model

is also shown.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.16 Variation of bond slip at cross sectional location I1 at 1.45, 1.68 and 1.9 in of

vertical displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.17 Variation of bond stress at cross sectional location I1 at 1.45, 1.68 and 1.9 in of


4.18 Variation of bond slip at cross sectional location I2 at 2.1, 2.35 and 2.58 in of


4.19 Variation of bond stress at cross sectional location I2 at 2.13, 2.35 and 2.58 in

of vertical displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.20 Designation of beam sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.21 Deformation [in] of SEC 2 at vertical displacements of 1.0 in (top) and 1.9 in

(bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.22 Deformation [in] of SEC 3 at vertical displacements of 1.0 in (top) and 1.9 in

(bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.23 Deformation [in] of SEC 1 at vertical displacement of 1.0 in . . . . . . . . . . 50




4.27 Moment at the boundary of the center column vs. imposed vertical displacement 52

4.28 Moment at the boundary of the side support vs. imposed vertical displacement . 53

4.29 Shear force response vs. imposed vertical displacement . . . . . . . . . . . . . 53

4.30 Variation of steel longitudinal strain εs,zz at cross sectional location S2 at 0.15,

0.3 and 0.48 in of vertical displacement (horizontal dashed line represents yield

strain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.31 Variation of concrete longitudinal strain εc,zz at cross sectional location C1 at

0.15, 0.3 and 0.48 in of vertical displacement (horizontal dashed lines represent

peak compressive strain and crack strain) . . . . . . . . . . . . . . . . . . . . 55




LIST OF FIGURES vii



strain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56









strain) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58




4.38 Variation of steel longitudinal strain εs,zz at cross sectional location S1 at about


yield strains) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


1.68 and 1.9 in of vertical displacement (horizontal dashed lines represent yield

strains) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.1 Analytical and experimental strain history of bottom rebar at the face of the

center column (SEC 4) vs. imposed vertical displacement . . . . . . . . . . . . 60

5.2 Analytical and experimental strain history of top rebar at the face of the side

support (SEC 1) vs. imposed vertical displacement . . . . . . . . . . . . . . . 62

5.3 Analytical and experimental beam rotations over 8 in from the face of center

column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4 Analytical and experimental beam rotations over 8 in from the face of the side

support over a length of 8 in . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

List of Tables3.1 Material parameters and characteristic values of uniaxial stress-strain relation-

ships of longitudinal and transverse steel reinforcement . . . . . . . . . . . . . 9

3.2 Material parameters and characteristic values of uniaxial compressive stress-

strain relationship of concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Parameters of tensile stress-strain relationship for reinforced concrete . . . . . 20

3.4 Initial characteristic values of the bond stress-slip relationships in region A and B 30

3.5 Characteristic values of interface bond stress-slip relationships of region A and

B as implemented in the finite element model . . . . . . . . . . . . . . . . . . 33

4.1 Analysis load steps and associated vertical displacement at the face of center

column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Values of steel reinforcement pull-out and push-in at about 5.26 in of vertical

displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

ACKNOWLEDGMENTS

I would like to express my gratitude to my advisor, Professor Mehrdad Sasani, for his

guidance, patience and invaluable support throughout this study. I would like to thank

Professor Dionisio Bernal for kindly accepting to read my thesis. I would also like to

thank Professor Peter Furth who supported my aspiration to study at Northeastern

University. I also wish to express my appreciation to my parents for their support and

encouragement.

.

Chapter 1 Introduction

1.1 Overview

Understanding the behavior of reinforced concrete (RC) beams subjected to large deformations

requires further analytical and experimental studies. In [1], analytical and experimental studies

of an RC beam subjected to large deformations are presented using a detailed 3-D finite element

model. Yet, the bond slip between steel reinforcement was not modeled. In this study, a

different finite element model of an RC beam is developed, incorporating bond slip.

1.2 Objectives

The goals of this thesis are:

1) To develop a detailed three dimensional finite element model of a RC beam to be subjected

to large deformations.

2) To evaluate the model by comparing the analytical results with the previously obtained

experimental data.

1.3 Structure of the Thesis

In chapter 2, the spatial discretization of the RC beam is discussed. The topology of the el-

ements used to model the concrete core and concrete cover, longitudinal and transverse steel

reinforcement as well as the interface between concrete and steel are introduced.

In chapter 3, the constitutive modeling of concrete, steel reinforcement and interface steel-

concrete is discussed. Using von Mises yield criteria, input stress-strain relationships for the

transverse and longitudinal steel reinforcement are introduced. Utilizing the Modified Com-

pression Field Theory, material modeling for concrete in compression and tension is presented.

Evaluating existing bond-slip relationships and the corresponding experimental data, input

bond-slip relationships for different regions in the RC beam are presented.

INTRODUCTION 2

In chapter 4, the results of the analysis are presented. The variation of stresses and strains

in reinforcement steel and concrete as well as of bond stresses and slippage along the beam

is shown. The deformations of the RC beam sections are discussed and pull-out and push-in

values of the anchored steel reinforcement are presented. The response of the RC beam at ma-

terial, section and element level is presented.

In chapter 5, analytical steel reinforcement strains are evaluated and compared with the corre-

sponding experimentally results. The analytical beam end rotations are also compared with the

experimental results.

In chapter 6, concluding remarks are presented.

Chapter 2 Spatial Discretization and Element

Topology

Figs. 2.1 to 2.3 show an RC beam reported in [1] and its cross sections. In order to capture

the three dimensional state of stress and strain in the RC beam, a 3-D finite element model is

developed. The interface between concrete and steel reinforcement is explicitly modeled using

interface elements.

Figure 2.1: View of the RC beam reported in [1]

Figure 2.2: Cross section A-A Figure 2.3: Cross section B-B

SPATIAL DISCRETIZATION AND ELEMENT TOPOLOGY 4

2.1 Cross Section Discretization

The discretization of beam section A-A is shown in Fig. 2.4. Concrete cover and core main

areas are built of quadrilateral surfaces. The steel reinforcement cross sections comprise oc-

tagonal surfaces. The transition region from octagonal to quadrilateral surfaces is discretized

using triangular surfaces. Given the symmetric cross section of the beam, only half of the beam

cross section is modeled.

Figure 2.4: Discretization of the cross section of the RC beam reported in [1]

2.2 3-D Discretization

In order to save computational time, only a quarter of the RC beam is modeled. That is, one half

of the cross section and one half of the two span beam is modeled. The finite element model

of the experiment reported in [1] is composed of 5 Segments (see Figs. 2.1, 2.2, 2.3 and 2.5).

Segment 1 includes a portion of the beam having the cross section shown in Fig. 2.3. Segments

2 and 3 have the cross section shown in Fig. 2.2. Segment 4 is the side support of the RC beam

and its right hand side boundary nodes are fixed in all three translational degrees of freedom.

Segment 5 is the center column part of the RC beam. At the left hand side boundary of segment

5, the experimental measured RC beam deformations will be imposed to the finite element

model. Transverse reinforcement is modeled using the embedded reinforcement concept which

is described later in this chapter. The top longitudinal reinforcement of the tested RC beam was

anchored in the side support using a 90 degree hook. The top reinforcement of the finite element

3-D DISCRETIZATION 5

model of the RC beam is anchored straight over a length of 11.0 in within the side support part

in order to minimize discretization effort. Only small values of steel reinforcement longitudinal

stress and strain are present at that location. Thus, it is believed that omitting to model the

anchorage hook in the model will not significantly influence the results of the computation.

20-node isoparametric brick elements are selected to model the concrete cover and core (see

Fig. 2.6). The longitudinal steel reinforcement of the RC beam is also modeled using 20-node

isoparametric brick elements. This was primarily because of the need for an element to be

compatible with the available interface elements in program DIANA 9.2 as discussed in the

next section.

X

Y

Z

Figure 2.5: 3-D finite element model of the RC beam reported in [1]

1

2

3 4

5

678

9

10

11

1213 14

15

16 17

1819

20

ξ

η

ζ

Figure 2.6: Topology of solid element

SPATIAL DISCRETIZATION AND ELEMENT TOPOLOGY 6

2.2.1 Interface between Longitudinal Reinforcement and Concrete

The area between longitudinal steel reinforcement and concrete is modeled using 16-node plane

quadrilateral isoparametric elements providing the capability to discretize curved surfaces (Fig.

2.7). The element comprises two overlaying planes. Each pair of nodes have coincident coor-

dinates which implies zero thickness of the interface element.

1 2

3

45

67

8

x

yz9 10

11

1213

1415

16

ux

uyuz

Figure 2.7: Topology of interface element and associated nodal displacements

The set of variables describing the element state of deformation and the element forces contains

nodal displacements ue, relative displacements Δu as well as normal traction tx and shear

tractions ty and tz. The location of the element nodes and the variables of the chosen interface

element are shown in Fig.2.7.

ue =

⎛⎜⎝

ux

uy

uz

⎞⎟⎠ t =

⎛⎜⎝

tx

ty

tz

⎞⎟⎠ Δu =

⎛⎜⎝

Δux

Δuy

Δuz

⎞⎟⎠ (2.1)

2.2.2 Transverse Reinforcement

Transverse reinforcement is modeled using embedded reinforcement. The strain of the rein-

forcement bar is computed from the displacement field of the concrete element in which the

transverse reinforcement is embedded.

Chapter 3 Material Constitutive Models

3.1 Steel Reinforcement

3.1.1 Theoretical Description

The use of solid elements to discretize the longitudinal steel reinforcement of the RC beam

(Section 2.2) requires a yield criteria to define the elastic limit of the steel reinforcement under

combined state of stresses. The DIANA 9.2 program provides the Tresca and von Mises yield

criteria which are both isotropic and independent of hydrostatic pressure and therefore suitable

for modeling steel reinforcement represented by means of solid elements. Experimental tests

on ductile steel have shown a particulary good agreement with von Mises criteria in describing

the onset of yielding not only under one-dimensional state of stress but also under combined

state of stresses [2]. Thus, the von Mises yield criteria is used in this study.

The von Mises yield criteria states that the onset of yielding depends on whether the octahedral

shear stress or the maximum distortional energy has reached a critical value [3]

f(σij, κ) = J2 − κ2 = 0 (3.1)

where κ represents the yield stress in pure shear; σij represents the state of stress and J2 de-

notes the second deviatoric stress invariant.

In the literature, Eq. 3.1 is also referred to as yield surface in principal stress space [3]. If the

von Mises criteria is calibrated to agree with the uniaxial tension yield stress σyield, κ is

κ =σyield√

3(3.2)

In program DIANA 9.2, the von Mises yield criteria is implemented such that it agrees with the

uniaxial stress-strain relationship of steel reinforcement. Eq. 3.3 represents the von Mises yield

surface in principal stress space calibrated to uniaxial tension test as it is available in program

DIANA 9.2. It can be obtained by substituting Eq. 3.2 into Eq. 3.1.

MATERIAL CONSTITUTIVE MODELS 8

f(σij, σyield) =√

3J2 − σyield = 0 (3.3)

If the von Mises yield criteria in Eq. 3.3 is used to describe steel reinforcement material behav-

ior, only elastic-perfectly plastic material behavior can be modeled since plastic deformation

are assumed to occur under a constant yield stress. Yet, steel reinforcement exhibits hardening

behavior. In order to model this post-yield response, a hardening rule is used within the theory

of incremental plastic flow. A hardening rule specifies the configuration of subsequent yield

surfaces in stress space if a material is loaded beyond the initial yield surface represented by

Eq. 3.3. Hardening rules can be categorized in isotropic, kinematic and mixed hardening rules.

In Program DIANA 9.2 an isotropic hardening rule is implemented in conjunction with the

von Mises yield criteria. It is adopted in this study to model the hardening behavior of steel

reinforcement and it is described subsequently.

Eq. 3.4 represents the von Mises yield criteria presented in Eq. 3.3 extended by an isotropic

hardening rule [4].

f(σij, σyield, εp) =√

3J2 − σyield(εp) = 0 (3.4)

εp (effective strain [3]), is an increasing function of the accumulated plastic strain increments

dεpij . There are two basic approaches within the theory of plastic flow to define the effective

strain εp, which are work and strain hardening approaches. In the Diana 9.2 program both

approaches are implemented. Yet, in case of von Mises isotropic hardening material, both lead

to the same scalar function for εp [3]. Within the strain hardening approach, the effective strain

can be defined as a suitable combination of plastic strain increments dεpij [3]. In the DIANA

9.2 program the following definition is implemented.

εp =

∫ √2

3dεp

ijdεpij (3.5)

Having εp, σyield(εp) can be found using uniaxial material test results. The corresponding

associated flow rule implemented in the program DIANA 9.2 is used in this study.

STEEL REINFORCEMENT 9

3.1.2 Material Parameters of Steel Reinforcement

Experimental test results [1] as well as the data provided in [5] are used to determine the ma-

terial parameters for longitudinal and transverse reinforcements, which are presented in Table

3.1. The modulus of elasticity E is taken from [5]. The yield stress σyield for both reinforce-

ments has been determined from tension tests conducted in [1]. For the longitudinal steel

reinforcement of the RC beam a trilinear idealization of the experimental uniaxial stress-strain

relationship is adopted. For transverse reinforcement a bilinear idealization of the experimental

uniaxial stress-strain relationship is used. Both uniaxial stress-strain relationships are shown in

Fig. 3.1.

Table 3.1: Material parameters and characteristic values of uniaxial stress-strain relationships

of longitudinal and transverse steel reinforcement

Longitudinal Reinforcement Transverse Reinforcement

E [ksi] 29,000 29,000

σyield [ksi] 75 60

εyield 0.0026 0.0021

Epostyield [ksi] 287 306

μ 0.3 0.3

σult = [ksi] 105 90

εult 0.13 0.13

Figure 3.1: Uniaxial stress-strain relationships of longitudinal and transverse steel reinforce-

ments


3.2 Concrete

In program DIANA 9.2, the three approaches to establish constitutive relations for concrete in

compression are: plasticity, elasticity and damage based approach.

Three-dimensional material models based on plasticity approach, namely the Drucker-Prager

model and the Modified Mohr-Coulomb model, are available in program DIANA 9.2. Within

the plasticity approach the total concrete strain εc is decomposed in an elastic part εpc and in a

plastic part εpc . The concept of strain decomposition appeals the most to the physical nature of

cracked concrete especially if combined with the smeared crack approach where the total strain

is decomposed to the ”crack strain” and the strain of the solid concrete [6]. Yet, preliminary

simulations using Drucker-Prager model in compression combined with a multi-directional

smeared crack model in tension revealed numerical difficulties (convergence problems). Ac-

cording to the manual of program DIANA 9.2 [4], such difficulties are likely to occur when in

one integration point simultaneously concrete compressive softening is accompanied by sev-

eral active cracks in the remaining lateral directions.

An elasticity based concept is the Modified Compression Field Theory. It was originally

developed within a two-dimensional framework [7]. The purpose was to predict the load-

deformation response of plane cracked reinforced concrete elements subjected to in-plane shear

and normal stresses. Subsequently, the concept was extended and enhanced, respectively, in or-

der to describe the load-deformation characteristic of reinforced concrete solids subjected to a

general three-dimensional state of stress [8]. This enhanced three-dimensional model con-

stitutes the basis for the elasticity based total strain concrete material model available in the

program DIANA 9.2.

Lastly, a damage based material model, namely the Modified Maekawa Model, is implemented

in the program DIANA 9.2. This model is based on two-dimensional and three-dimensional

cyclic loading data. The damage based approach was not considered in this study.

3.2.1 2-D Modified Compression Field Theory

Using a total strain based approach, concrete behavior in compression and tension is modeled

in this study. The total strain based approach for concrete material modeling in the program DI-

ANA 9.2 is based on the Modified Compression Field Theory [7]. In the Modified Compression

CONCRETE 11

Field Theory, stress-strain relationships are evaluated in terms of average stresses and strains.

Figs. 3.2 and 3.3 show a reinforced concrete element and the corresponding two-dimensional

states of stress and strain.

Figure 3.2: 2-D state of stress in a re-

inforced concrete element [7]

Figure 3.3: 2-d state of strain in a re-

inforced element [7]

In order to relate the principal average stresses σc,1 and σc,2 to principal average strains ε1 and

ε2 , an average stress-strain constitutive relationship is required. Such an average stress-strain

relationship can differ from local stress-strain relationships determined from standard material

tests.

An extensive series of reinforced concrete panels were tested in [7] in order to obtain infor-

mation with regard to characteristics of such an concrete average stress-strain constitutive re-

lationship. For details with regard to the set up the reader is referred to [7]. In these tests,

in-plane normal stresses σx , σy and shear stresses τxy, depicted in Fig. 3.2, were applied to the

reinforced concrete panels. The corresponding average strains εx , εy and γxy, shown in Fig.

3.3, were measured. Based on these measured quantities, the following average stress-strain

constitutive relationship for concrete in compression was developed:

σc2 = σc2,max ·[2

(ε2

ε′c

)−(

ε2

ε′c

)2]

(3.6)

where σc2 represents the second principal stress in concrete (maximum compression); ε2

represents the second principal strain in concrete and


σc2,max

f ′c

=1

0.8 − 0.34 · ε1/ε′c(3.7)

where f ′c represents the concrete compressive strength; ε′c represents the concrete strain at f ′

c

and ε1 represents the first principal strain in concrete.

Eq. 3.6 and 3.7 reflects the main findings of the study with regard to the compressive load-

deformation characteristics of reinforced concrete panels. As one can see, looking at the con-

stitutive relationships presented above, the principal average compressive stress σc2 is not only

a function of the principal compressive strain ε2 but it also depends on the principal tensile

strain ε1. In other words, if cracked concrete is subjected to high tensile strains normal to the

compressive direction, it shows softer and weaker characteristic in compression.

3.2.2 3-D Modified Compression Field Theory

Section 3.2.1 was focussed on the load-deformation response of plane reinforced concrete el-

ements subjected to plane state of stress and strain incorporating compressive strength degra-

dation of concrete if tensile strain in the lateral direction prevails. However, the enhancement

of compressive strength and ductility of concrete due to confining effects has not been consid-

ered. The 3-D extension of the Modified Compression Field Theory [8] and the corresponding

proposed orthotropic secant material matrix for a concrete solid element, summarized in the

following section, accounts for the aforementioned concrete material behavior.

The three dimensional secant material matrix of an orthotropic concrete material in principal

directions is [8]

[D] =1

φ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ec1(1 − υ32υ23) Ec1(υ12 + υ13υ32) Ec1(υ13 + υ12υ23) 0 0 0

Ec2(υ21 + υ31υ23) Ec2(1 − υ31υ13) Ec2(υ23 + υ21υ13) 0 0 0

Ec3(υ31 + υ21υ32) Ec3(υ32 + υ12υ31) Ec3(1 − υ21υ12) 0 0 0

0 0 0 φGc12 0 0

0 0 0 0 φGc23 0

0 0 0 0 0 φGc31

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3.8)

where

φ = 1 − υ23υ32 − υ21υ12 − υ31υ13 − υ21υ32υ13 − υ31υ12υ23 (3.9)

CONCRETE 13

The term υij stands for the i-th strain component due to a stress in the j-th direction or simply

Poison’s ratios. The shear moduli are given [8]

Gc12 =Ec1Ec2

Ec1(1 + υ12) + Ec2(1 + υ12)(3.10)

Gc23 =Ec2Ec3

Ec2(1 + υ23) + Ec3(1 + υ32)(3.11)

Gc13 =Ec1Ec3

Ec1(1 + υ13) + Ec3(1 + υ31)(3.12)

where Ec1, Ec2 and Ec3 represent secant elastic moduli of concrete.

For the definition of the secant elastic moduli the reader is referred to [8]. Within a simulation,

the state of stress and strain is transformed into the corresponding principal axis system where

the uniaxial stress-strain relationships are evaluated.

Concrete Constitutive Laws within the 3-D Modified Compression Field Theory

Within the 3-D Modified Compression Field Theory, an uniaxial compressive stress-strain re-

lationship based on the work in [9] is implemented. It reads [8]

σc3 = −fpεc3

εp

n

n − 1 + (εc3/εp)nk(3.13)

where σc3 represents the third principal stress in concrete (compression); εc3 represents the third

principal strain in concrete; fp represents the uniaxial peak compressive stress of concrete; εp

represents the strain in the concrete at fp; n = 0.80; k = 1 for the ascending branch of Eq.

3.13 and k = 0.67 +fp

62for the descending branch of Eq. 3.13.

The uniaxial peak stress fp as well strain εp at peak stress increase if confining stresses exist. In

order to account for confinement effects, a failure criteria is implemented in the 3-D Modified

Compression Field Theory [8]

f = 2.0108J2

f ′c

+ 0.9714

√J2

f ′c

+ 9.1412σc1

f ′c

+ 0.2312I1

f ′c

− 1 = 0 (3.14)


where I1 represents the first invariant of the stress tensor; J2 represent the second deviatoric

stress invariant; σc1 represents the first principal stress in concrete and f ′c represents the com-

pressive strength of concrete.

Using the failure criteria Eq. 3.14, the compressive stress σc3,f that causes failure in presence

of stresses σc1, σc2 can be found. The stress σc3,f is used to compute the peak stress factor

Kσ =σc3,f

f ′c

(3.15)

which is used to modify the uniaxial peak strength fp of concrete (Eq. 3.13).

fp = Kσf ′c (3.16)

Due to confinement effects, concrete shows, along with an enhanced uniaxial peak compressive

stress fp, an increasing strain εp at fp. This behavior is captured by mean of the peak strain

factor Kε. For low confining stresses (Kσ < 3):

Kε = 0.2036K4σ − 2.819K3

σ + 13.313K2σ − 24.42Kσ + 13.718

√Kσ + 1 (3.17)

and for larger confining stresses:

Kε = 5Kσ − 4 (3.18)

The strain εp in Eq. 3.13 is then modified:

εp = ε0

{Kσ

(1 − σc3

σc3,f

)+ Kε(

σc3

σc3,f

)}(3.19)

where σc3 represents the current stress in the third principal direction and with the peak strain

under uniaxial compression and the strain at peak stress under uniaxial compression is

ε0 =n

n − 1

f ′c

Ec(3.20)

where n = 0.80 and Ec represents the concrete modulus of elasticity.

CONCRETE 15

3.2.3 Concrete in Tension

Cracks in concrete can be modeled using discrete or smeared cracks. Within the discrete crack

approach concrete cracking is modeled by means of a displacement discontinuity at the in-

terface between concrete elements. The smeared crack concept takes a different approach in

which the cracked material is assumed to be a continuum and the effect of cracking is described

by means of an appropriate tensile stress-strain relationship. The crack is therefore smeared out

over the effected elements. A smeared crack model will be used in this study to model cracking

of concrete.

Within the original smeared crack concept implemented in program DIANA 9.2, the total strain

of a cracked concrete solid ε is decomposed into a part εcr of the crack and a part εco of the

solid material. This allows for implementing constitutive laws incorporating crack dilatancy

which is essentially a coupling between the crack normal direction n and tangential directions

s, t [6]. However, using the total strain based material model for concrete in compression it

is conceptually not possible to combine the approach of decomposed strain in tension with

the total strain approach of concrete in compression. Therefore, a total strain based smeared

crack model [4], available in program DIANA 9.2, is used to model concrete in tension. The

tensile strength of concrete fct is deployed as conditional detection. Once a crack is initiated,

two distinctive approaches as how to handle the crack subsequently exist: the single rotating

crack approach and the single fixed crack approach. In program DIANA 9.2, both approaches

are implemented within the context of total strain based concrete constitutive modeling. In the

single fixed crack approach, the orientation of the crack remains unaltered during the entire

computation process. In the total strain based material models, the stress-strain relationships

of concrete are evaluated in principal stress space. Thus, a disadvantage arises when using

the total strain single fixed crack model. At incipient of cracking, the element principal axes

of strain are replaced by crack directions. Subsequently, the uniaxial constitutive stress-strain

relationships are evaluated in a coordinate system constrained to the crack direction. This yield

a misalignment of the principal directions of stresses and the principal directions of strains in

subsequent load steps. This can lead to spurious high tensile stresses that might exceed the

tensile strength fct multiple times [6]. The misalignment of principal strain and crack direction

can be avoided if a rotating crack model is used. The orientation of the crack rotates with the

axes of principal strain. In conjunction with a tensile cut-off criteria, tensile stresses which

exceed the tensile strength of concrete are then avoided. Thus, the rotating crack model is used

in this study.


The post cracking response of reinforced concrete differs from that of plain concrete. While

plain concrete is a brittle material which exhibits softening behavior upon crack initiation if

subjected to uniaxial tensile stress, concrete in RC structures continues to carry tensile stress

between the cracks due to the transfer of forces from the tensile reinforcement to the concrete

through bond. In order to include this ”tension stiffening” effect a material model proposed in

[10] is used. The affiliated material parameters and tensile stress-strain relationship is presented

in section 3.2.4.

3.2.4 Concrete Material Parameters

Concrete Compressive Stress-Strain Relationships

In the side support, center column and the RC beam cover, the compressive relationship pre-

sented in Eq. 3.13 is adopted. The compressive cylinder strength f ′c , listed in table 3.2, has

been determined in compression tests [1]. The modulus of elasticity for concrete is estimated

using

Ec = 57, 000√

f ′c [psi] (3.21)

The strain at peak stress under uniaxial compression ε0 is determined using Eq. 3.20. The

resulting uniaxial stress-strain relationship is shown in Fig. 3.4.

Table 3.2: Material parameters and characteristic values of uniaxial compressive stress-strain

relationship of concrete

Region Ec [ksi] ν f′c [ksi] ε0 εu

Core Concrete 4400 0.15 6.6 0.0022 0.02

Rest of Specimen 4400 0.15 6.0 0.002 0.006

The total strain based material model, as it is implemented in program DIANA 9.2, proved to

be non suitable for the core elements. The approach to modify the peak compressive stress fp if

confining stresses exist, taken in the 3-D Modified Compression Field Theory (Section 3.2.2),

has been completely adopted. Yet, the strain at peak compressive stress εp is modified using

a different approach. The peak strain factor Kε (Eq. 3.17 and 3.18) is set to be equal to peak

stress factor Kσ (Eq. 3.15). However, setting Kε = Kσ is suitable for unconfined concrete

or the cover concrete of the RC beam [8]. For confined concrete, the ratio of strain at peak

compressive stress εp to strain at peak stress under uniaxial compression ε0 increases much

CONCRETE 17

Figure 3.4: Uniaxial compressive stress-strain relationship for concrete cover, side support and

center column

faster than the ratio of peak stress fp to compressive strength of concrete f ′c [8]. Furthermore,

for the enhanced ductility of confined concrete is accounted for using

σc3 = −fp

(1 − (1 − r)

εc3 − εp

εu − εp

)(3.22)

where r represents a factor which models the residual strength and εu denotes the ultimate strain

of concrete

εu =

(fp

fcc

)γ

εp (3.23)

and γ = 3.0.

The ultimate strain εu, predicted using Eq. 3.23, does not lead to sufficient ductility of the

concrete core in the presence of the amount of confinement as is provided by the transverse re-

inforcement of the RC beam. This can be shown by comparing the existing model in program

DIANA 9.2 with Manders model [11], summarized below.


The uniaxial compressive stress strain relationship developed in [11] is

σc3 =f ′

ccxr

r − 1 + xr(3.24)

with

x =εc3

εcc

(3.25)

and

εcc = εc

[1 + 5

(f ′

cc

f ′c0

− 1

)](3.26)

where f ′c0 represents the compressive strength of unconfined concrete; εc0 represents the corre-

sponding concrete strain (taken to 0.002); f ′cc represents the compressive strength of confined

concrete and εcc represents the corresponding concrete strain.

In order to estimate the compressive strength f ′cc of confined concrete, the five parameter

William-Warnke failure surface is used [11].

f ′cc = f ′

c0

(− 1.2456 + 2.254

√1 +

7.94f ′l

f ′c0

− 2f ′

l

f ′c0

)(3.27)

where f ′l represents the effective lateral confining stress.

The effective lateral confining stress is a function of the yield stress fyh, of the transverse

reinforcement, the confinement effectiveness coefficient ke and the ratio ρ of the volume of

transverse confining steel to the volume of the RC beam core.

f ′l = keρfyh (3.28)

The confinement effectiveness coefficient is

ke =Ae

Acc

(3.29)

with

Ae =

(1 −

n∑i=1

(w′i)

2

6bcdc

)(1 − s′

2bc

)(1 − s′

2dc

)(3.30)

CONCRETE 19

Acc = Ac(1 − ρcc) (3.31)

where Ae represents the area [in2] of the effectively confined concrete core; Acc represents the

area [in2] of the core of the RC beam enclosed by the center lines of the perimeter hoop and ρcc

represents the ratio of the area of the longitudinal reinforcement to the area of the core of the

RC beam section.

In Fig. 3.5, the cross section of the RC beam at the face of the center column is depicted. The

height of the compressive zone at peak compressive stress is estimated to a third of the section

height. The strain at the top of the core section is assumed to be at εco = 0.002.

Figure 3.5: Effectively confined concrete core

The resulting uniaxial compressive stress-strain relation of the concrete core and the trilinear

approximation, which is used in the finite element model, are shown in Fig. 3.6. The material

parameters for the concrete core and the characteristic values of the uniaxial compressive stress-

strain relationship are listed in Table 3.2.

Concrete Tensile Stress-Strain Relationship

A rotating crack model, outlined in section 3.2.3, is adopted in order to incorporate cracking

of the concrete in the RC beam. In order to account for tension stiffening effects of reinforced

concrete in tension, which contributes to the overall flexural stiffness of the beam, a piecewise

linear tensile stress-strain relationship developed in [10] is adopted. The model is depicted in

Fig. 3.7 and the characteristic values are listed in Table 3.3. The tensile strength used as tension

cut-off condition is

f ′ct = 6.45 ·

√f ′

c [psi] (3.32)


Figure 3.6: Uniaxial compressive stress-strain relationship for concrete core [11] and the tri-

linear approximation

Figure 3.7: Uniaxial tensile stress-strain relationship of reinforced concrete [10]

Table 3.3: Parameters of tensile stress-strain relationship for reinforced concrete

Pt Rt St Ft εcr fct [Ksi ]

0.8 0.45 4 10 1.4e-4 0.5

INTERFACE BETWEEN CONCRETE AND STEEL REINFORCEMENT 21

3.3 Interface between Concrete and Steel Reinforcement

In order to capture the contribution of bond to the flexibility and deformation capacity of the

beam, the interface between concrete and steel reinforcement elements is explicitly modeled

by means of three-dimensional interface elements.

3.3.1 Mechanics of Interaction between Steel Reinforcement and Con-

crete

In RC elements, at the interface between the rebars and concrete, three mechanisms are acti-

vated if the rebars are subjected to tensile or compressive stresses: chemical adhesion, mechan-

ical interaction between the rebar ribs and the concrete keys as well as friction [12]. Fig. 3.8

schematically shows bond stress-slip relationships [13].

For bond stress values τ ≤ τI , the concrete surrounding the rebar remains in the elastic stage

and chemical adhesion governs the transfer of the rebar stress into the surrounding concrete.

The slip between the concrete and the rebar remains small [13].

Splitting

Failure

Splitting induced

Pull-Out Failure

Pull-OutFailure

Confinement

0.5 f’ c

�I I

III

IV a

IV b

Av. BondStress

Slippage s

Curve 1 Curve 2 Curve 3�I

II

Figure 3.8: Qualitative local bond stress-slip relationships [13]

For bond stress values τ > τI , the capacity of the chemical adhesion to transfer the rebar stress

into the concrete is exceeded and bearing stresses in front of the rebar ribs are generated (me-

chanical interaction). Transverse cracks, shown in Fig. 3.9, arise in the surrounding concrete


causing the rebar to slip. As the bond stress τ increases, the concrete subjected to bearing

stresses crushes, shown in Fig. 3.9, and the bond stiffness decreases (Fig. 3.8, Stage II).

T e n s io n

C o m p re s s io n

R e b a r

T ra n s v e rs eM ic ro c ra c k s

c ru s h e dC o n c re te

Figure 3.9: Transverse cracks originated at the tips of the steel reinforcement ribs and crushed

concrete in front of the ribs of the rebar [12]

The outward component of the bond stress is balanced by ring tensile stresses [14], shown in

Fig. 3.10. If the tensile stresses become large enough, longitudinal cracks develop along the

rebar. If no or an insufficient amount of transverse reinforcement is provided, the longitudinal

cracks reach the outer concrete surface and bond failure occurs. (Fig. 3.8, Stage III, Curve 1).

As the amount of transverse reinforcement increases, the spreading of the longitudinal cracks

is increasingly restricted as shown in Fig 3.11. The longitudinal cracks do not reach the outer

concrete surface and as a result, larger bond stresses can develop [13] (Fig. 3.8, Stage IVa,

Curve 2). The development of longitudinal cracks remains within an small area around the

rebar section if the confinement action, either provided by transverse reinforcement or trans-

verse pressure, further increases. More transverse cracks in the surrounding concrete arise and

the crushing of the concrete in front of the rebar ribs continues [12]. This bond mechanism

provides the largest bond strength (Fig. 3.8, stage IVb, Curve 3). The magnitude of bond stress

in stage IVb can be as high as 0.5f ′c.

At maximum bond stress τmax, the concrete keys between the rebar ribs start to shear off. With

increasing slip, an increasingly larger part of the concrete keys are sheared off and the bond

stress decreases until only the frictional part of the bond resistance τresidual remains when the

concrete surface is smoothed out [12] (Fig. 3.8, Curve 2 and 3).


Figure 3.10: Radial component of

bond stress balanced by tensile ring

stresses in the surrounding concrete

[14]

Figure 3.11: Radial cracks induced by

tensile ring stresses and transverse re-

inforcement action [13]

3.3.2 Review of Existing Bond Stress-Slip Relationships

In [12], pull-out tests have been conducted using a specimen aimed to simulate conditions

within a beam-column joint. The specimen consisted of a single deformed rebar embedded

in a concrete block over a length of le = 5db. The bond stress along the embedment length

was assumed to be evenly distributed due to the short anchorage length. For the majority of

the tests, the concrete block was reinforced with longitudinal (perpendicular to the axis of the

anchored rebar) and transverse (parallel to the axis of the anchored rebar) steel reinforcement.

For the remaining tests the concrete block was not reinforced in order to obtain reference data

for unconfined concrete. One protruding end of the anchored rebar was subjected to load under

displacement control while the slip was measured as the movement of the unloaded end of the

rebar with respect to the concrete anchorage block. The load applied to the rebar was reacted

as a compressive force at the face of the concrete block. Bond stress-slip relationships were

deduced by taking applied forces at given slip value and converting them into bond stress using

Eq. 3.33.

τ =F

π · db · le (3.33)

where F represents the applied force; db represents the rebar diameter and le represents the

embedment length.


The test program consisted of several series each designed to investigate a single varying pa-

rameter that influences bond characteristic while all other specimen parameter were kept con-

stant. The longitudinal reinforcement varied from #2 to #8 rebars and the transverse reinforce-

ment varied from #2 to #4 stirrups. Furthermore, the diameter of the anchored rebar varied

from 0.5 in to 1.0 in in order to obtain information with regard to the influence of the rebar di-

ameter on the bond characteristics. Additionally, a test series was conducted where the specific

rib area varied from fr= 0.066 to fr= 0.12. The specific rib area is

fr =Ar

π · db · sr

(3.34)

where Ar represents the area of the projection of a single rib on the cross-section of the rebar;

sr represents the rib spacing and dbrepresents the rebar diameter.

A test series with concrete compressive strength varying from f ′c=4350 psi to f ′

c=7975 psi was

also conducted. In all tests, a splitting crack developed in the plane of the longitudinal axis of

the anchored rebar and the failure of bond was observed if the concrete was not confined by

reinforcement. The concrete between the ribs of the rebar was intact. In case of confined con-

crete, the growth of the splitting crack was controlled by the longitudinal reinforcement. The

load could be increased further and failure of bond was caused by pulling out of the anchored

rebar. It is reported that the concrete between the rebar ribs was completely sheared off and

almost pulverized. As a result of these tests, Eq. 3.35 was proposed to represent an average of

experimentally obtained bond stress-slip curves for confined concrete.

τ = τ1 ·(

s

s1

)α

; s ≤ s1 (3.35)

where τ represents the bond stress; s represents the slip of the rebar; τ1 = 13.5N/mm2;

s1 = 1 mm and α = 0.4

According to [12], bond between rebars and concrete scatters. The values for τ1 and α varied

between τ1 ≈ 11.5N/mm2 to τ1 ≈ 15.5 N/mm2 and α ≈ 0.33 to α ≈ 0.45. It is reported that

the influence of the the rebar diameter on bond characteristics was rather small. The values of

τmax increased approximately proportional to√

f′c and the corresponding slip values decreased

almost proportional to 1√f′c

. Varying the clear spacing between the rebars from a minimum

value of sb = 1db to a maximum value of sb = 4db resulted in an increase of τmax of about

20%. It is furthermore reported that the distance between the rebar ribs sr greatly influences


the characteristic of bond. The slip value s1 at which τmax was reached increased if sr was

enlarged. Thus, modification factors are proposed which may be applied to Eq. 3.35 if it is

intended to use the bond stress-slip relationship (Eq. 3.35) in specific cases where different

confining reinforcement details, rebar diameter, deformation pattern and concrete compressive

strength prevail. These modification factors are presented and applied later in this chapter.

In [15], three direct tension tests were conducted. The test set up was designed to simulate

conditions prevailing in a region of a beam between flexural tensile cracks. The specimen con-

sisted of a single #8 rebar centrally embedded over a length of 18db in a cross section concrete

block of 5x5x18 in. The rebar was subjected to tensile forces at the projecting ends. A bond-

stress slip relationship was determined using a different technique compared to aforementioned

pull-out test. The rebar was equipped with internal strain gages which were distributed along

the embedment length. Additionally strain gages were embedded in the concrete block at a

distance of 0.25 in from the rebar surface. The concrete strain and the strain of the embedded

rebar were measured and recorded at different locations along the embedment length. Subse-

quently, each strain function was integrated resulting in the displacement of the concrete and

the rebar. The slip was then computed by subtracting the concrete displacements from the re-

bar displacements. The findings of this study show that the bond stress-slip relationship varies

significantly along the embedment length. The maximum bond stress varied from 4√

f ′c at a

distance of 2 in from the loaded end of the rebar to 10√

f ′c at a distance of 6 in from the loaded

end of the rebar. The slip at which τmax was reached varied from 0.0005 in at 2 in to 0.0015 in

at 6 in from the loaded end of the rebar. Using the test results, Eq. 3.36 was proposed in [16]

to represent a best fit to the average of the experimental results.

τ = f ′c · (16.7s − 8260s2 + 1.12 · 106s3) [ksi] (3.36)

where f ′c represents concrete compressive strength and s represents the rebar slip.

In [17], a series of direct tension tests on concrete square prisms was conducted. Each specimen

was axially reinforced with one #8 central rebar. The rebar was instrumented with internal

strain gages. This technique allowed for obtaining the steel stress distribution along the length

of the rebar. The length of the concrete prism was 16 in which yield an embedment length of

16db. Cross sectional dimensions varied from 2x2 in to 6x6 in which led to different ratios of

concrete cover to rebar diameter. The specific rib area fr for all rebars was nearly constant at

fr=0.14 to 0.15. The concrete compressive strength varied from f ′c = 4300-5000 psi. The slip


of the rebar was measured at 0.25 in from the rebar surface using a micrometer. Bond stress-slip

relationships were deduced by using Eq. 3.33. The force F in Eq. 3.33 was derived as force

transferred into the concrete using information provided by the stress distribution along the

rebar. It is reported that in all tests no crushing of concrete in front of the rebar lugs occurred.

Therefore, it is concluded that the observed slip is due to the internal cracking of the first layer

of the concrete surrounding the rebars (see Fig. 3.15) and due to bending and/or cracking of the

concrete keys near the rebar ribs. Furthermore, it was observed that the concrete compressive

strength had an insignificant effect on measured slip values. Yet, the maximum bond stress was

observed to be proportional to√

fc. It is also reported that with increasing cover thickness, the

slip at which maximum bond stress was reached decreased owing to the enhanced restraining

capacity of the concrete mass. As a result of this study, Eq. 3.37 was proposed to represent

a best fit to experimentally obtained data which were normalized to a concrete compressive

strength of f ′c = 5000 psi.

τ = 1.95 · 106s − 2.35 · 109s2 + 1.39 · 1012s3 − 0.33 · 1015s4 [psi] (3.37)

where f ′c represents concrete compressive strength and s represents the slip of the rebar.

In Fig. 3.12 the bond stress-slip relationships are shown for two slip ranges. As can be seen,

the bond stress-slip relationships differ considerably. For very small values of up to a slip of

about 0.0001 in the bond stiffness is rather close. But with increasing slip values the difference

in the tangent modulus is apparent. According to [12], a large scatter in the initial bond stiff-

ness may be caused by inaccuracies in measuring the slip between the rebar and the concrete

correctly. If slip values are deduced from measured concrete and rebar strains, as conducted in

[15], the error even accumulates. Also, the bond stress-slip relationship for rebars embedded

over a length > 5db cannot be considered as constant along the embedment length. It varies

considerably at distinct locations along the rebar. Furthermore, it is reported that the position

of the rebars during casting influences the initial bond stiffness [12]. Rebars cast horizontally

show much smaller initial bond stiffness compared to rebars cast vertically.


Figure 3.12: Bond stress-slip relationships [12], [15] and [17]

Another potential source of the scatter in initial bond stiffness is the use of different test speci-

mens with different stress conditions prevailing in the concrete surrounding the rebar. While in

pull-out tests the concrete is subjected to longitudinal compression, in direct tension tests the

concrete is subjected to tensile stresses. Experimental findings indicate that one of the main

source of scatter in initial bond stiffness is the use of rebars with different deformation pattern

which lead to different specific rib area fr for identical rebar diameter [12]. Larger specific rib

area yield larger bearing area of the rebar ribs which in turn induces smaller bearing stresses

on the concrete between the rebar ribs at equal rebar stress level. This reduces the strain of

the concrete in the vicinity of the rebar which lead to smaller slip values at the steel-concrete

interface. Additionally, it is reported that for identical specimens maximum bond stress τmax

was reached at slip values of smax= 0.028 in and smax=0.054 in for rebars having a specific rib

area of fr = 0.12 and fr = 0.066, respectively [12]. This corresponds to an increase in initial

bond stiffness of about 50% due to an increase in specific rib area of about 50%. The specific

rib area reported in [12] and [17] varied from fr = 0.065 to fr = 0.14 − 0.15, respectively,

whereas the diameter of the rebar in both test series was the same.

3.3.3 Parameters influencing Bond Performance

Geometry and Deformation Pattern of Steel Reinforcement

As summarized and discussed in the previous section, the rib geometry of the rebar has paramount

importance among the other parameters that effect bond behavior. It has been found that the

maximum bond stress τmax strongly depends on the specific rib area fr. Experimental results

[12] indicate that the slip smax at which τmax is reached or the initial bond stiffness, respectively,


increases as the specific rib area increases. Furthermore, according to [12], the rib spacing

sr has a significant effect on the characteristic values of a bond stress-slip relationship. With

increasing values of sr, the slip smax at which maximum bond stress τmax is reached, increases.

Similarly, the slip at which the bond stress-slip relationship levels off to the frictional bond

stress τresidual, increase.Another important factor is the cover thickness. With an increasing

cover thickness, the restraining effect on the rebar is enhanced [17]. The level of stress, present

in the rebar, also effects the bond performance. As long as the rebar remains within the linear

elastic range, the influence of steel reinforcement stress remains small. Yet, experimental tests

show that yielding of the steel reinforcement has a negative effect on bond. It results in a sharp

nonlinear descending branch in the bond stress-slip relationship once yielding of the rebar has

occurred [13].

Concrete Characteristics

The maximum bond stress τmax varies with increasing concrete compressive strength f ′c [12].

It has been experimentally shown that τmax is proportional to√

f ′c [12] and [17]. This is

because bond action results from the localized pressure induced into the concrete in front of

the rebar ribs and the pressure is directly related to the shear component of bond stress [13].

Furthermore, experimental results indicate that slip values corresponding to τmax decrease ap-

proximately proportional to 1/√

f ′c which results in an increasing bond stiffness [12].

Confinement Effects

Transverse compressive stresses favor bond action independent on whether they result from

active or passive confinement. Active confinement, resulting from a direct support or from a

continuity of a column in a beam-column joint, is more efficient to prevent splitting failure

since it does not depend on the bond stress itself. On the contrary, passive confinement devel-

oped by concrete cover and transverse reinforcement is less effective since it has its origin in

the dilatancy of concrete cracks and the development of cracks is related to bond stress [13].

Passive confinement controls the spreading of the longitudinal splitting cracks and prevents

pure splitting failure, as reported in [12]. Experimental tests show that if sufficient passive

confinement is provided or the stirrup confinement index Ω is within a certain range, maximum

bond strength increases to a certain extent [13].

Additional factors influencing bond characteristic are the clear rebar spacing sb as well as the

rebar diameter. In [12], a decreasing maximum bond stress is reported if the clear rebar spacing

sb falls below 4db. A slight decrease of the maximum bond stress of about 10% for increasing


rebar diameter and equal values of relative rib area fr was observed in [12]. Furthermore, rebar

corrosion, rusting and the loading also affect bond-slip [13].

3.3.4 Implemented Bond Stress-Slip Relationships

The constitutive law, available in program DIANA 9.2, which governs the behavior of the 3-

dimensional interface elements is [8]

(τn

τt

)=

⎛⎝kn 0

0∂τ(st)

∂st

⎞⎠ ·

(Δsn

Δst

)(3.38)

where τn, τt represent the normal and the tangential component of the interface traction or the

bond stress; τ(st) represents the bond stress-slip relationship provided as input; kn represents

the normal stiffness of the interface between concrete and rebar and sn, st represent the normal

and the shear component of the interface relative displacement or the bond slip.

As can be seen, the shear and normal components are decoupled. Thus, the interaction between

the shear component of bond and the lateral contraction and extension of the rebars due to Poi-

son’s effect cannot be modeled. Furthermore, no communication protocol between interface

elements and concrete elements or steel reinforcement elements is implemented in program

DIANA 9.2. Thus, bond deterioration due to reinforcement yielding and concrete crushing

cannot be considered.

In the finite element model of the RC beam, two distinctive regions with regard to bond con-

ditions are defined. Region A (Fig. 3.13) is defined well as confined. Given the existence of

longitudinal and transverse reinforcement, it is assumed that the bond characteristic in region

A is of a pull-out failure type [13]. The bond stress-slip relationship for the interface elements

around the rebars subjected to compressive stress and for the interface elements around the

rebars subjected to tensile stress are assumed to be equal which complies with experimental

findings [12]. Table 3.4 provides the values of τmax, τresidual and of s1, s2 and s3 in region A,

which are adopted according to the test data obtained in [12]. Note that the pattern of the bond

stress-slip relationship in both regions is the same. The pattern was proposed in [12] and is

shown in Fig. 3.14 along with the designation of the characteristic points of the curve. The

concrete compressive strength f ′c , the clear bar spacing sb and the rebar diameter db differ from

those prevailing in the pull-out tests [12]. Furthermore, the deformation pattern of the rebars

used in [1] is different from that of the rebars used in [12].


Region B

Region ARegion A S 3S 2S 1 Slip s

m ax

BondStress

res idua l

τ

τ

Figure 3.13: Defined regions of bond

in the finite element model

Figure 3.14: Pattern of the bond

stress-slip relationship in region A

and B

Table 3.4: Initial characteristic values of the bond stress-slip relationships in region A and B

Region τmax [psi] τresidual [psi] s1 [in] s2 [in] s3 [in]

A 1640 410 0.035 0.07 0.4

B 800 200 0.01 0.02 0.1

Thus, the initial values of the characteristic points of the bond stress-slip relationship in region

A, given in Table 3.4, are subsequently modified according to suggestions made in [12] in order

to account for differences in aforementioned parameters.

Region B within the finite element model of the RC beam is defined. As discussed earlier, the

values of τmax deduced from direct tension tests are considerable smaller than the values for

τmax obtained in pull-out test in the confined concrete blocks. A bond model presented in [14]

is used to approximate the initial value of τmax in region B. In the model, a cracked concrete

sleeve around the rebar is restraint by an outer solid sleeve, subjected to tensile hoop stresses,

such that the rebar does not slide out of the concrete [14] (Fig. 3.15). The initial value of τmax

is estimated using Eq. 3.39. The value of τmax represents the bond stress at the incident of

bond failure of a embedded rebar once the thickness of the inner sleeve reaches a critical value,

namely 1.664 db (Fig. 3.15). The transverse reinforcement of the tested RC beam comprised

with a diameter of 0.135 in placed at 2.7 in. This is considerably less than in region A. Thus, the

restraining effect on the concrete surrounding the rebars is not as pronounced as in region A.

However, it provides to some extent additional restraining effects yielding a descending branch

of the bond stress-slip relationship and an increased maximum bond stress τmax.


τmax = fct ·cy +

db

21.664 · db

= 800 psi (3.39)

where

cy = 0.8175 in concrete cover of the RC beam

db = 0.375 in rebar diameter

fct = 500 psi concrete tensile strength

Figure 3.15: Hydraulic pressure anal-

ogy taken from [14]

The initial value of τmax = 800 psi compares reasonable to experimental data presented in sec-

tion 3.3.2 (Fig. 3.12), given the scatter in maximum bond stress. With regard to the initial

values of slip in region B, data given in [15] and [17] vary significantly as presented earlier

and do not seem to be reliable. Therefore, the initial value of s1 in region B is adopted based

on a pull-out test on concrete specimen without confining reinforcement [12]. Values for the

slip s2 and s3 are also suggested. However, these values are based on data which were obtained

in pull-out tests showing splitting failure of the specimen yielding a sharp descending branch

of the bond stress-slip relationship (in Fig. 3.14 s2 = s1). Yet, due to the confinement effect,

exerted by the transverse reinforcement of the RC beam, the pattern of the bond stress-slip

relationship in region B is equal to that of region A. Thus, the ratio of s2 to s1 and of s3 to s1 in

region A, which computes to 2 and 11.5, respectively, is used in order to obtain initial values

for s2 and s3 in region B. Following the same approach, the ratio of τmax to τresidual in region A,

which computes to 4, is used to obtain an initial value of τresidual in region B.

Modification of Initial Characteristic Values due to Concrete Compressive Strength

The compressive strength of the concrete of the RC beam is f ′c = 6 ksi. The initial value for

τmax in region A is based on pull-out tests in specimens having a concrete compressive strength

of f ′c = 4.35 ksi. In order to convert the initial values to account for the concrete strength of

the RC beam, τmax and τresidual in region A are increased by a factor of about√

64.35

= 1.17


[12]. In region B, τmax is not modified since Eq. 3.39 was developed based on pull-out tests in

specimens having also a compressive strength of f’c = 6 ksi. The initial value for s1 in region

A and B is reduced by a factor of 11.17

based on experimental findings in [12].

Modification of Initial Characteristic Values due to Confining Reinforcement

In [12], the ratio

∑Asv∑As

(3.40)

where Asv represents the area of the confining reinforcement and As represents the area of the

anchored RC beam reinforcement, varied between 0.0 to 4.0. Eq. 3.40 gives a value of about

1.6 for the bottom reinforcement of the RC beam and a value of about 4.0 for the top reinforce-

ment. These values compare favorable to the values in [12]. Thus, the initial characteristic

values of τmax and τresidual in region A are not modified.

The stirrup confinement index Ω is [13]

Ω =Ast

A∗ (3.41)

where Ast represents the area of the transverse reinforcement; A∗ = n · db · Δz represents the

area of the rebars in the splitting plane; n represents the number of rebars; db represents the

rebar diameter and Δz = 2.7 in represent the distance at which the transverse reinforcement is

placed along the RC beam (Fig. 2.5, Page 3).

For the RC beam, Ω is 0.006 and 0.015 for five #3 rebars (top of section) and two #3 rebars

(bottom of section). Experimental results indicate that for Ω between 0.006 to 0.014 a maxi-

mum bond stress τmax of 0.12 f ′c to 0.18f ′

c can be expected [13]. This yield a maximum bond

stress of τmax ≈ 720−1080 psi considering that f ′c = 6000 psi. The initially estimated value of

τmax = 800 psi represents a lower bound of maximum bond stress [14]. Thus, the initial char-

acteristic value of τmax in region B is increased by 200 psi in order to account for confinement

effects exerted by the transverse reinforcement of the RC beam.

Modification of Initial Characteristic Values due to Deformation Pattern

The specific rib area fR of the #3 rebars used in [1] is 0.1. fR of the rebars used in [12] is 0.11.

Thus, no modification factor is applied to τmax in region A.


The rib spacing sr of the rebars used in [1] is sr = 0.2 in. Yet, the rib spacing of the rebars

used in [12] is sr = 0.4 in. Thus, the initial characteristic values for s1, s2 and s3 in region A

and B are modified by a factor of 0.2 in0.4 in = 0.5, as suggested in [12].

Modification of Initial Characteristic Values due to Rebar Spacing

In the RC beam, the clear rebar spacing sb is 0.9 in for the top rebars and 4.27 in for the bottom

rebars. The ratio

sb

db

(3.42)

where db represents the rebar diameter,is computed to be 2.4 and 11.38 for the top and the

bottom rebars, respectively. Experimental findings suggest a reduction of τmax and τresidual by

10% for values of 2.4 [12]. This reduction is applied to the initial values in region A and B.

The final values for the characteristic points of the bond stress-slip relationships, which are im-

plemented in the finite element model, are listed in Table 3.5 and Fig. 3.16 shows the affiliated

bond stress-slip relationships.

Table 3.5: Characteristic values of interface bond stress-slip relationships of region A and B as

implemented in the finite element model

Region τmax [psi] τresidual [psi] s1 [in] s2 [in] s3 [in]

A 1740 500 0.016 0.035 0.2

B 1000 250 0.004 0.008 0.05


Figure 3.16: Bond stress-slip relationship of interface elements in region A and B as imple-

mented in the finite element model

Chapter 4 Analytical ResultsIn this chapter, the variation of stress, strain, bond stress, and slippage along the beam are pre-

sented. Fig. 4.2 shows the locations where the results for concrete, steel and interface elements

were obtained. Contour plots of two longitudinal profiles are also presented. Longitudinal

profile 1 includes core and cover concrete whereas longitudinal profile 2 includes only cover

concrete, as depicted in Fig. 4.1. The results in this chapter are presented for different load

steps (LS). The vertical displacement associated with each load step is listed in Table 4.1.

Figure 4.1: Designation of longitudi-

nal views used for contour plots

Figure 4.2: Locations of concrete C,

steel S and interface I elements

Table 4.1: Analysis load steps and associated vertical displacement at the face of center column

Load Step (LS) 1 2 3 4 5 6 7 8 9

Vertical Displacement [in] 0.15 0.3 0.48 0.76 1.0 1.2 1.45 1.68 1.9

Load Step (LS) 10 11 12 13 14 15 16 17

Vertical Displacement [in] 2.13 2.35 2.58 2.8 3.21 3.62 4.44 5.26

ANALYTICAL RESULTS 36

4.1 Analytical Results at Material Level

In this section, analytical results along the beam are presented. Note that the center line of the

two-span RC beam (where the vertical displacement is applied) is located at zero distance (see

Fig. 2.5). The face of the side support is at a distance of 82 in.

Concrete in Tension

Behavior of concrete in tension is modeled using rotating cracks (Sec. 3.2.3). Thus, at every

location within the beam, tensile stresses are expected to be less than the tensile strength of

fct = 500 psi. In Fig. 4.3, the variation of concrete longitudinal stress σc,zz along the beam

at location C3 is shown for three load steps LS1, LS2 and LS3 (see Table 4.1). As can be

seen, longitudinal tensile stresses σc,zz do not exceed the tensile strength fct = 500 psi at

any location. The distance between location of peak tensile stress of 500 psi and the face of

the center column in Fig. 4.3 can be considered as the length over which the concrete tensile

strain has exceeded the crack strain of εcr = 0.00014 and the concrete elements go through

the descending branch of uniaxial tensile stress-strain relationship (see Fig. 3.7). The tensile

stress equal to zero indicates that concrete tensile strain has exceeded ultimate tensile strain

εu,cr = 0.0014 (see Fig. 3.7). Within the distance between the peak tensile stress of 500 psi

and inflection point (zero bending moment), concrete elements are in the pre-cracking elastic

stage. As can be seen in Fig. 4.4, for larger deformations, tensile stresses no longer reach the

tensile strength of fct = 500 psi. This indicates that all concrete elements between the face of

the center column and inflection point and top bar cut-off location at C3 have passed the crack

strain of εcr = 0.00014. Three contour plots of concrete first principal stress σc,1, shown in

Fig. 4.5, also confirm that stress values have not exceeded the tensile strength of fct = 500 psi

throughout the analysis.

ANALYTICAL RESULTS AT MATERIAL LEVEL 37

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−3500

−3000

−2500

−2000

−1500

−1000

−500

0

500

DISTANCE ALONG BEAM [in]

AV

ER

AG

E S

TR

ES

S [p

si]

CONCRETE CORE BOT MIDDLE STRESS OVER LS

LS1LS2LS3

Faceof

CenterColumn

Faceof

SideSupport

Top Bar Cut−OffLocation


Figure 4.3: Variation of concrete longitudinal stress σc,zz at cross sectional location C3 at 0.15,

0.3 and 0.48 in of vertical displacement

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−6,000

−5,000

−4,000

−3,000

−2,000

−1,000

0

500


AV

ER

AG

E S

TR

ES

S [p

si]


LS7LS8LS9

Faceof

CenterColumn

Faceof

SideSupport






-.2E4-.1E4-5000250500

-.2E4-.1E4-5000250500

-.2E4-.1E4-5000250500

Figure 4.5: Longitudinal profile 1 of first principal stress σc,1 [psi] at three vertical displace-

ments: 0.76 (Top); 1.0 (Middle) and 1.2 in (Bottom)

Concrete in Compression

Figs. 4.6 and 4.7 show the variation of concrete longitudinal stress σc,zz along the beam at cross

sectional locations C1 and C2 (see Fig. 4.2), respectively. Maximum concrete compressive

stress of σc,zz = -5227 psi (Fig. 4.6) and σc,zz = -5789 psi (Fig. 4.7) are calculated to occur

at 1.9 in of vertical displacement for location C1 and C2, respectively. For concrete elements

at cross sectional location C3 and C4, maximum concrete compressive stress in longitudinal

direction of σc,zz = -5911 psi (Fig. 4.8) and σc,zz = -6278 psi (Fig. 4.9) are calculated at a

vertical displacement of 1.2 in, respectively. The peak compressive strengths of f ′c= 6000 psi

for cover concrete and f ′c=6600 psi for core concrete have not been exceeded throughout the

analysis. With regards to third principal stress, maximum values of σc,3 = 6000 psi and σc,3 =

6600 psi for cover concrete and core concrete, respectively, are calculated (Figs. 4.10 and

4.11). This is consistent with the Modified Compression Field Theory where the state of stress

is evaluated in principal stress space. Values for the third principal stress larger than the strength

of uniaxial compressive stress-strain relationship are not expected to occur.


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90-5500

-5000

-4500

-4000

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500


AV

ER

AG

E S

TRE

SS

[psi

]

CONCRETE COVER TOP MIDDLE STRESS OVER LS

LS7LS8LS9

Faceof

SideSupport

Faceof

CenterColumn

Top BarCut-Off

Location

BarCut-Off

Location



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90-6000

-5500

-5000

-4500

-4000

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500


AV

ER

AG

E S

TRE

SS

[psi

]

CONCRETE CORE TOP MIDDLE STRESS OVER LS

LS7LS8LS9

Faceof

CenterColumn

Faceof

SideSupport

Top BarCut-Off

Location

Top BarCut-Off

Location




0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−6000

−5500

−5000

−4500

−4000

−3500

−3000

−2500

−2000

−1500

−1000

−500

0

500


AV

ER

AG

E S

TR

ES

S [p

si]

CONCRETE COVER BOT MIDDLE STRESS OVER LS

LS4LS5LS6

Top BarCut−OffLocation


Faceof

SideSupportFace

ofCenter

Column



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−6500

−6000

−5500

−5000

−4500

−4000

−3500

−3000

−2500

−2000

−1500

−1000

−500

0

500


AV

ER

AG

E S

TR

ES

S [p

si]


LS4LS5LS6

Faceof

SideSupport

Faceof

CenterColumn






-.66E4-.6E4-.4E4-.2E40500

-.66E4-.6E4-.4E4-.2E40500

-.66E4-.6E4-.4E4-.2E40500

-.66E4-.6E4-.4E4-.2E40500

Figure 4.10: Longitudinal profile 1 of third principal stress σc,3 at 1.2, 1.45, 1.68 and 1.9 in

(plots from top to bottom) of vertical displacement (the face of the side support is located on

the left, the face of the center column is located on the right)

-.66E4-.6E4-.4E4-.2E40500

-.66E4-.6E4-.4E4-.2E40500

-.66E4-.6E4-.4E4-.2E40500

-.66E4-.6E4-.4E4-.2E40500

Figure 4.11: Longitudinal profile 2 of third principal stress σc,3 at 1.2, 1.45, 1.68 and 1.9 in

(plots from top to bottom) of vertical displacement (the face of the side support is located on

the right, the face of the center column is located on the left)


Steel Reinforcement

Figs. 4.12 and 4.13 show the variation of longitudinal strain εs,zz and stress σs,zz , respectively,

for steel reinforcement elements at cross sectional location S1 (see Fig. 4.2). At the vicinity

of the side support, a maximum tensile strain of εs,zz = 0.0341 is calculated at a vertical dis-

placement of 5.26 in (LS17). At the vicinity of the center column, a maximum compressive

strain of εs,zz = -0.0355 is computed at the same level of vertical displacement. Maximum

steel reinforcement stress values in longitudinal direction of σs,zz = 92300 psi in tension and

σs,zz = -94500 psi in compression are calculated at a vertical displacement of 5.26 in.

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04


AV

ER

AG

E E

LEM

EN

T S

TR

AIN

S1 Strain Variation

LS16

LS17


Faceof

SideSupport


Faceof

CenterColumn

Figure 4.12: Variation of steel longitudinal strain at cross sectional location S1 at 4.44 and 5.26

in of vertical displacement (horizontal dashed lines represent yield strains)

Fig. 4.14 shows the variation of longitudinal strain εs,zz for steel reinforcement elements at

cross sectional location S2 (see Fig. 4.2). At a vertical displacement of 5.26 in, a maximum

tensile strain of εs,zz = 0.13 is computed at the vicinity of the center column and a maximum

compressive strain of εs,zz = -0.0586 at the vicinity of the side support. Fig. 4.15 shows the

tensile stress-strain relationships of the first three steel reinforcement elements from the face of

the center column. As can be seen, the bar fracture occurs in the second element at a vertical

displacement of 5.26 in.


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−95,000

−80,000

−60,000

−40,000

−20,000

0

20,000

40,000

60,000

80,000

95,000


AV

ER

AG

E E

LEM

EN

T S

TR

ES

S [p

si]

S1 Stress Variation

LS16LS17

Faceof

SideSupport


Faceof

CenterColumn


Figure 4.13: Variation of steel longitudinal stress at cross sectional location S1 at 4.44 in and

5.26 in of vertical displacement (horizontal dashed lines represent yield stresses)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14


AV

ER

AG

E E

LEM

EN

T S

TR

AIN

S2 Longitudinal Strain Variation

LS16LS17


Faceof

SideSupport

Faceof

CenterColumn


Figure 4.14: Variation of steel longitudinal strain at cross sectional location S2 at 4.44 and 5.26

in of vertical displacement (horizontal dashed lines represent yield strains)


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.140

1

2

3

4

5

6

7

8

9

10

11x 10

4

STRAIN

ST

RE

SS

[psi

]

STEEL STRESS−STRAIN

1st Element2nd Element3rd ElementInput Uniaxial Stress−Strain

Figure 4.15: Tensile stress-strain of the first three steel reinforcement elements from the face

of the center column at cross sectional location S2. (Uniaxial stress-strain relationship of steel

reinforcement as implemented in the finite element model is also shown.)

Interface between Concrete and Steel Reinforcement

Figs. 4.16 and 4.17 show the variation of bond stress and bond slip ,respectively, for interface

elements at location I1 (see Fig. 4.2). At the face of the center column a change in sign of bond

stress and bond slip values, compared to the remaining interface elements along the beam, is

observed. Steel reinforcement in compression is pushed into the center block. In order to sat-

isfy the compatibility, the first element of the beam steel reinforcement has to move into the

center block yielding the same sign of bond stress and bond slip values as for interface elements

within the center column. After a transition, bond stress and bond slip variation is consistent

with the change in bending moment along the beam. The same pattern of bond stress and bond

slip variation can be observed for interface elements around steel reinforcement in tension in

the vicinity of the side support. As opposed to steel reinforcement in compression, the an-

chored steel reinforcement in tension is pulled out of the side support. Again, for compatibility

reasons, bond stress and bond slip values of interface elements around the first steel reinforce-

ment element of the beam have the same sign as bond stress and bond slip values of interface

elements within the side support. The maximum bond stress of τmax = 1000 psi is calculated

at a vertical displacement of about 1.68 in at the vicinity of the center column.


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−8.5

−7.5

−6.5

−5.5

−4.5

−3.5

−2.5

−1.5

−0.5

0.5

1.5

2.5

3.5

4.55

x 10−3


AV

ER

. ELE

M. R

EL.

DIS

PLA

CE

ME

NT

[in]

I1 Slip Distribution

LS7LS8LS9

Faceof

CenterColumn

Faceof

SideSupport



Figure 4.16: Variation of bond slip at cross sectional location I1 at 1.45, 1.68 and 1.9 in of

vertical displacement

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−1500

−1000

−500

0

500

1000

1300


AV

ER

AG

E E

LEM

EN

T T

RA

CT

ION

[psi

]

I1 Bond Stress Distribution

LS7LS8LS9

Faceof

CenterColumn


Faceof

SideSupport


Figure 4.17: Variation of bond stress at cross sectional location I1 at 1.45, 1.68 and 1.9 in of



The corresponding bond slip is larger than the yield slip of s1= 0.004 which is consistent with

the implemented bond stress-slip relationship. Bond stress and bond slip values for interface

elements further from the face of the center column remain below maximum bond stress τmax

and yield slip s1 throughout the analysis.

Similar results are obtained for the variation of bond stress and bond slip (see Figs. 4.18 and

4.19) for interface elements at cross sectional location I2 (see Fig. 4.2). As for interface

elements at cross sectional location I1 (see Fig. 4.2), the maximum bond stress of τmax =

1000 psi is calculated only for interface elements around compressive reinforcement at the

vicinity of the face of the side support (Fig. 4.19). Also, the corresponding bond slip is larger

than 0.004 which is again consistent with the implemented bond stress-slip relationship. Bond

stress and bond slip values for interface elements more far from the side support remain below

maximum bond stress τmax and yield slip s1, respectively, throughout the analysis. However,

as opposed to interface elements at cross sectional location I1, the maximum bond stress τmax

is calculated to occur at a vertical displacement of about 2.13 in.

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015


AV

ER

. ELE

M. R

EL.

DIS

PLA

CE

ME

NT

[in]

I2 Slip Distribution

LS10LS11LS12

Faceof

SideSupport



Faceof

CenterColumn

Figure 4.18: Variation of bond slip at cross sectional location I2 at 2.1, 2.35 and 2.58 in of


BOND SLIP AND SECTION DEFORMATION 47

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−1750

−1500

−1000

−500

0

500

1000

1500


AV

ER

AG

E E

LEM

EN

T T

RA

CT

ION

[psi

]

I2 Bond Stress Distribution

LS10LS11LS12



Faceof

CenterColumn

Faceof

SideSupport

Figure 4.19: Variation of bond stress at cross sectional location I2 at 2.13, 2.35 and 2.58 in of


4.2 Bond Slip and Section Deformation

In this section, computed nodal displacements at selected locations will be evaluated with re-

gard to sectional deformation as well as steel reinforcement pull-out and push-in. Fig. 4.20

shows the cross sections along the beam for which vertical lines, V2 and V3, are defined.

Along these lines nodal displacements were obtained from the program. The designations of

sections, for instance SEC 1 V2, are subsequently used in this section.

Faceof

CenterColumn

Faceof

SideSupport

8.0in 8.0in

SEC 4 SEC 3 SEC 2 SEC 1 V2V3

Figure 4.20: Designation of beam sections

Interface elements were implemented in the finite element model. As a result, the anchored

steel reinforcement is either pulled out (tension) or pushed into (compression) the side support

block or the center column block, respectively. In table 4.2, the calculated maximum values of

steel reinforcement pull-out and push-in at SEC 1 and SEC 4 are listed.


Table 4.2: Values of steel reinforcement pull-out and push-in at about 5.26 in of vertical dis-

placement

SEC1 SEC4

V2 V3 V2 V3

Pull Out 0.027 in 0.0136 in 0.04 in -

Push In 0.0167 in - 0.0169 in 0.013 in

The maximum value of bar pull-out is calculated for steel reinforcement in tension anchored in

the center column block (SEC 4 V2). The calculated maximum value of bar pull-out at SEC

1 V2 is 32% less. A comparison of pull-out values calculated for SEC 1 V2 and SEC 1 V3

shows a difference of about 50%. The calculated bar push-in values are within similar range at

both beam end sections.

Figs. 4.21 and 4.22 show nodal displacements at SEC 2 V2 and SEC 3 V2 at a vertical dis-

placement of 1.0 in and 1.9 in. Note that the horizontal axis represents the nodal displacements

in longitudinal direction (z-direction) and the vertical axis represents the section height. As can

be seen, calculated nodal displacements indicate that during the deformation, cross sections re-

main plain if only the nodal displacements of concrete elements are considered. However, as

a result of the bond slip, the cross sections do not remain plain if the nodal displacements of

steel reinforcement elements are also considered.

Figs. 4.23 and 4.24 show nodal displacements at SEC 1 V2 and SEC 4 V2 at a vertical dis-

placement of 1.0 in and Figs. 4.25 and 4.26 show nodal displacements at SEC 1 V2 and SEC

4 V2 at a vertical displacement of 1.9 in. Similar to SEC 2 V2 and SEC 3 V2, the calculated

nodal displacements of the concrete elements of SEC 4 V2 indicate that the section remains

plain throughout the analysis. Yet, as for SEC 2 V2 and SEC 3 V2, due to the bond slip cross

section SEC 4 does not remain plain. The concrete part of SEC 4 shows a non-plain deforma-

tion due to the highly localized pressure at the face of the side support induced by the single

compressive rebar.

BOND SLIP AND SECTION DEFORMATION 49

1

2

3

4

5

6

7

8

-0.01 0 0.01

1

2

3

4

5

6

7

8

-0.01 0 0.01

1

2

3

4

5

6

7

8

-0.01 0 0.01

1

2

3

4

5

6

7

8

-0.01 0 0.01

Figure 4.21: Deformation [in] of SEC

2 at vertical displacements of 1.0 in

(top) and 1.9 in (bottom)


3 at vertical displacements of 1.0 in

(top) and 1.9 in (bottom)


1

2

3

4

5

6

7

8

-0.01 0 0.01

Model: 05_LC21: LoadStep: 5 LNodal TDTXMax/Min onYmax = 7.5Xmax = .84VariationS1_V2


1 at vertical displacement of 1.0 in

1

2

3

4

5

6

7

8

-0.01 0 0.01


4 at vertical displacement of 1.0 in

1

2

3

4

5

6

7

8

-0.02 -0.01 0 0.01 0.02

Model: 05_23_08_3LC3: Load case 3Step: 9 LOAD: 1Nodal TDTX...G TDTZMax/Min on whole graph:Ymax = 7.5 Ymin = 0Xmax = .139E-1 Xmin = -.144E-1Variation along a lineS1_V2

Figure 4.25: Deformation [in] of SEC 1 at vertical displacement of 1.9 in

ELEMENT END FORCES 51

1

2

3

4

5

6

7

8

-0.01 0 0.01 0.02

Figure 4.26: Deformation [in] of SEC 4 at vertical displacement of 1.9 in

4.3 Element End Forces

In this section, element end forces are presented. In order to calculate the beam internal forces,

nodal reaction forces were obtained at the boundaries of the finite element model (see Fig.

2.5). Figs. 4.27 and 4.28 show beam bending moment response at the boundary of the center

column block and at the boundary of side support block, respectively, versus the imposed

vertical displacement. The horizontal axis represents the vertical displacement imposed at the

boundary of the center column block. The vertical axis represents the moment calculated at the

boundary of the center column and at the boundary of the side support, respectively. The peak

positive bending moment at the boundary of the center column is 175.1 kip-in at a vertical

displacement of 1.9 in (Fig.4.27). The calculated response shows a constantly diminishing

slope up to 1.0 in of vertical displacement. Beyond this level of deformation, the slope of

the bending moment increases. At 1.9 in of vertical displacement the bending moment shows

a drop. Further increased boundary deformations result in decreasing values of the bending

moment. At a vertical displacement of 5.26 in a bending moment of 85.3 kip-in is computed.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

2

4

6

8

10

12

14

16

18x 10

4

Imposed Vertical Displacement [in]

Res

istiv

e M

omen

t [lb

in]

Moment at Center Column Boundary vs. Imposed Vertical Displacement

Figure 4.27: Moment at the boundary of the center column vs. imposed vertical displacement

The peak negative bending moment at the boundary of the side support is 249.9 kip-in at a ver-

tical displacement of 1.68 in (Fig. 4.28). As opposed to the bending moment at the boundary

of the center column, no intermediate increase of the slope of the bending moment is observed.

Instead, the slope of the bending moment is constantly diminishing up to the peak at 1.68 in

of vertical displacement. Increasing boundary deformation beyond vertical displacement of

1.68 in results in a negative slope of the bending moment. At a vertical displacement of 1.9

in, a sharp drop of the bending moment is observed. As boundary deformations are further

increased, the bending moment constantly decreases and a bending moment of 108.8 kip-in is

calculated at a vertical displacement of 5.26 in.

Fig. 4.29 shows the calculated shear force response of the beam. The vertical axis represents the

shear force of the beam and the horizontal axis represents the imposed vertical displacement.

The calculated peak shear force is 4.48 kips at a vertical displacement of 1.9 in. Beyond vertical

displacement of 1.9 in, the shear force reduces as imposed boundary deformations are increased

until maximum vertical displacement of 5.26 in is reached. This level of vertical deformation

corresponds to a shear force of 2.05 kips.

ELEMENT END FORCES 53

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0x 10

5


Mom

ent [

lbin

]

Moment at Side Support Boundary vs. Imposed Vertical Displacement

Figure 4.28: Moment at the boundary of the side support vs. imposed vertical displacement

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

500

1000

1500

2000

2500

3000

3500

4000

4500


She

ar F

orce

[lb]

Shear Force at SEC 1 vs. Imposed Vertical Displacement

Figure 4.29: Shear force response vs. imposed vertical displacement


4.4 State of Strain at Beam Ends

In this section, beam response at the section level is presented. Note that only results at SEC 1

(Face of Side Support) and SEC 4 (Face of Center Column) are discussed (see 4.20). Note that

strain values, used to describe the beam response at section level, are average element strains

of the first elements from either the face of the center column or the face of the side support.

Considering an element length of about 1.0 in, results were calculated at about 0.5 in from the

face of the side support or the the face of the center column. In figures presented, this location

is associated with the first marker (either from the face of the center column or from the face of

the side support) of the curves illustrating the variation of strains. Furthermore, cross sectional

locations defined in Fig. 4.2 are used in figure captions.

The bottom bar at SEC 4 (face of center column) shows a strain of about εs = 0.003 at a vertical

displacement of 0.48 in (see Fig. 4.30, LS3), which is larger than the yield strain. At the same

vertical displacement, the concrete at the top of SEC 4 is at a strain of εc = −0.002 (Fig. 4.31,

LS3)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5x 10

−3


AV

ER

AG

E L

ON

GIT

UD

INA

L S

TR

AIN

S2 Steel Longitudinal Strain Variation

LS1LS2LS3

Faceof

SideSupport

Faceof

CenterColumn



Figure 4.30: Variation of steel longitudinal strain εs,zz at cross sectional location S2 at 0.15, 0.3

and 0.48 in of vertical displacement (horizontal dashed line represents yield strain)

STATE OF STRAIN AT BEAM ENDS 55

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−4

−3

−2

−1

0

1

2

3

4

5

6x 10

−3


AV

ER

AG

E L

ON

GIT

UD

INA

L S

TR

AIN

C1 Concrete Longitudinal Strain Variation

LS1LS2LS3


Faceof

SideSupport


Faceof

CenterColumn

Figure 4.31: Variation of concrete longitudinal strain εc,zz at cross sectional location C1 at 0.15,

0.3 and 0.48 in of vertical displacement (horizontal dashed lines represent peak compressive

strain and crack strain)

At a vertical displacement of 0.76 in, concrete at the bottom of SEC 1 (face of side support)

shows a strain of about εc = −0.0025 (Fig. 4.32, LS4). The top bar at SEC 1 is at a strain of

εs = 0.002 at this level of vertical displacement which indicates that it has not yielded (Fig.

4.33, LS4).

At 1.2 in of vertical displacement, concrete at the top of the core at SEC 4 is at a strain of

εc = −0.0022 (Fig. 4.34, LS6). Note that concrete at the top at SEC 4 is at a strain of

εc = −0.015 (Fig. 4.35, LS6) which indicates a significant variation of longitudinal strain over

the section height. The top bar at SEC 4 shows a strain of εs = −0.00145 (Fig. 4.33, LS6). The

bottom bar at SEC 4 is at a strain of εs = 0.025 (Fig. 4.36, LS6). Furthermore, at this level of

vertical displacement, concrete at the bottom of the core at SEC 1 is at a strain of εc = −0.0025

(Fig.4.37, LS6). Concrete at the bottom at SEC 1 shows a strain of εc = −0.008 (Fig. 4.32,

LS6). Concurrently, the top bar at SEC 1 has yielded with a strain of εs = 0.0043 (Fig. 4.33,

LS6). The bottom bar at SEC 1 is at a strain of εs = −0.0015 and has not yielded (Fig. 4.36,

LS6).


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−0.008

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05


AV

ER

AG

E L

ON

GIT

UD

INA

L S

TR

AIN


LS4LS5LS6



Faceof

SideSupport

Faceof

CenterColumn




0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−0.0015

−0.001

0

0.001

0.002

0.003

0.004

0.005


AV

ER

AG

E L

ON

GIT

UD

INA

L S

TR

AIN


LS4LS5LS6

Faceof

CenterColumn


Faceof

SideSupport





0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−0.0025−0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.015


AV

ER

AG

E L

ON

DIT

UD

INA

L S

TR

AIN


LS4LS5LS6

Faceof

SideSupport


Faceof

CenterColumn





0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−0.016−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.017


AV

ER

AG

E L

ON

GIT

UD

INA

L S

TR

AIN


LS4LS5LS6

Faceof

CenterColumn



Faceof

SideSupport





0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−0.002

0

0.005

0.01

0.015

0.02

0.025

0.03


AV

ER

AG

E L

ON

GIT

UD

INA

L S

TR

AIN

S2 Steel Longitudinal Strain Distribution

LS4LS5LS6

Faceof

SideSupport

Faceof

CenterColumn





0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−0.003

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035


AV

ER

AG

E L

ON

GIT

UD

INA

L S

TR

AIN


LS4LS5LS6

Faceof

SideSupport

Faceof

CenterColumn







The top bar at SEC 4 is yielded with a strain of εs = 0.0026 at a vertical displacement of 1.45

in (Fig. 4.38, LS7). Concrete at the top of the core at SEC 4 is at a strain of εc = −0.0057 at

this level of vertical displacement. At a vertical displacement of 1.68 in, the bottom bar at SEC

1 is yielded with a strain of εs = −0.0045 (Fig. 4.39, LS8).

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−0.009

−0.005

0

0.005

0.01

0.015

0.016


AV

ER

AG

E L

ON

GIT

UD

INA

L S

TR

AIN


LS7LS8LS9



Faceof

SideSupport

Faceof

CenterColumn

Figure 4.38: Variation of steel longitudinal strain εs,zz at cross sectional location S1 at about

1.45, 1.68 and 1.9 in of vertical displacement (horizontal dashed lines represent yield strains)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045


AV

ER

AG

E L

ON

GIT

UD

INA

L S

TR

AIN


LS7LS8LS9

Faceof

SideSupport

Faceof

CenterColumn



Figure 4.39: Variation of steel longitudinal strain εs,zz at cross sectional location S2 at 1.45,

1.68 and 1.9 in of vertical displacement (horizontal dashed lines represent yield strains)

Chapter 5 Comparison between Analytical and

Experimental ResultsIn the chapter, analytical and experimental results are compared. The comparison includes

strains of reinforcing bars and the beam end rotations.

5.1 Steel Reinforcement Strain

Fig. 5.1 compares analytical and experimental strains of the bottom rebar at the face of the cen-

ter column versus the imposed vertical displacement. Note that experimental data are available

only up to a strain of about 0.033.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

IMPOSED VERTICAL DISPLACEMENT [in]

ST

RA

IN

Analytical StrainExperimental Strain

Figure 5.1: Analytical and experimental strain history of bottom rebar at the face of the center

column (SEC 4) vs. imposed vertical displacement

The analytical strain reaches tensile yield strain of εs = 0.0025 at about 0.48 in of vertical

displacement. The experimental strain reaches tensile yield strain at about 0.68 in of vertical

displacement. After yielding, the slope of experimental strain changes drastically compared

STEEL REINFORCEMENT STRAIN 61

to the slope of analytical strain. This behavior is related to the yield plateau exhibited by the

uniaxial stress-strain relationship of the rebars used in the experimental study. In this region

of the uniaxial stress-strain relationship, strain increases with little or no increase in stress.

Therefore, material stiffness drops considerably leading to large strain increments as vertical

displacement is increased. The yield plateau is not modeled by the uniaxial stress-strain re-

lationship implemented in the model (see Fig. 3.1). This causes significantly different slopes

of the experimental and analytical tensile strains above the yield displacement. After the yield

plateau of the uniaxial stress-strain relationship, stress increases again with increasing strain

yielding enhanced material stiffness. This leads to decreasing tensile strain increments as the

vertical displacement is further increased. Beyond 0.78 in of vertical displacement the slope

of experimental tensile strain approaches the slope of analytical tensile strain. Beyond a ver-

tical displacement of about 1.05 in, the analytical tensile strain is in good agreement with the

experimentally measured tensile strain until a vertical displacement of about 1.6 in is reached.

Fig. 5.2 shows analytical and experimental tensile strains of the top rebar at the face of the

side support vs. the imposed vertical displacement. The analytical tensile strain reaches yield

strain of εs = 0.0025 at about 1.0 in of vertical displacement. The experimental tensile strain

reaches yield strain 1.5 in of vertical displacement. As for the tensile strain history at the face

of the center column, a diminished material stiffness (yield plateau of uniaxial stress-strain

relationship is reached) causes large tensile strain increments in the post yield range of the

experimental strain. After the yield plateau of uniaxial stress-strain relationship, the slope of

experimental tensile strain decreases and the strain increases linearly as vertical displacement is

increased and the analytical tensile strains approach experimental results as the imposed verti-

cal displacement increases. The analytical tensile strains at the face of the center column and at

the face of the side support reach the yield strain at a lower level of vertical displacement com-

pared to the vertical displacement at which experimental tensile strains reach yield strain. The

yield plateau of experimental uniaxial stress-strain relationship is not modeled by the stress-

strain relationship implemented in the finite element model. This leads to a difference between

the slope of the analytical and experimental strains at the vertical displacement level associated

with the yielding strain. However, both analytical tensile strain histories are in good agreement

with experimentally measured tensile strains.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.005

0.01

0.015

0.02

0.025

0.03


ST

RA

IN

Analytical StrainExperimental Strain

Bar Fracture(Analytical)

Bar Fracture(Experimental)

Figure 5.2: Analytical and experimental strain history of top rebar at the face of the side support

(SEC 1) vs. imposed vertical displacement

5.2 Beam End Rotations Fig. 5.3 the shows analytical and experimental beam end rotations measured from the face of

the center column over a length of 8 in. Except for the vertical displacements less than about

0.3 in, both experimental and analytical beam end rotations increase almost linearly. Analytical

beam end rotation is in good agreement with measured beam end rotation considering differ-

ence of only about 13% at about 5.26 in of vertical displacement.

Fig. 5.4 compares the analytical and experimental beam end rotations measured from the face

of the side support over a length of 8 in. Beyond a vertical displacement of about 1.0 in a

change in the slope of analytical beam end rotation can be observed. The change of slope in

the experimental beam end rotations starts later at about 1.6 in of vertical displacement. At

about 5.26 in of vertical displacement analytical beam end rotation exceeds experimental beam

end rotation by approximately 20%. As for beam end rotations at the face of center column,

the analytical beam end rotation at the face of the side support is in good agreement with

experimental beam end rotation.

63

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15


RO

TA

TIO

N [r

ad]

Analytical Rotation Face of Center ColumnExperimental Rotation Face of Center Column Bar Fracture

(Analytical)


Figure 5.3: Analytical and experimental beam rotations over 8 in from the face of center column

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065


RO

TA

TIO

N [r

ad]

Analytical Rotation Face of SupportExperimental Rotation Face of Support

Bar Fracture(Analytical)


Figure 5.4: Analytical and experimental beam rotations over 8 in from the face of the side

support over a length of 8 in

CONCLUDING REMARKS 64

Chapter 6 Concluding Remarks

The detailed finite element model of the continuous beam developed in this study allowed es-

timating analytical results that in general were in good agreement with the previously obtained

experimental responses of the beam subjected to large deformations. One of the important

response measures in large deformation response of RC beams is the deformation capacity cor-

responding to the first bar fracture. The analytical vertical displacement associated with the

first bar fracture underestimated the corresponding experiential value by about 12%.

The current version of the Diana program does not account for the interaction between the

concrete damage (crushing and cracking) and the bond deterioration. The inclusion of such

an interaction would extend the plastic zone, leading to a more ductile response, which in turn

would reduce the difference between the analytical and experimental vertical displacements

associated with the first bar fracture.

The element rotation is another important and practical measure of deformation response. The

analytical beam rotations over eight inches at the beam ends were in good agreement with the

experimental results. The finite element modeling described in this study can be used in esti-

mating rotation capacities of RC elements.

Bibliography[1] Sasani, M., and Kropelnicki, J. (2007)., ”Progressive Collapse Analysis of an RC Struc-

ture.”, The Structural Design of Tall and Special Buildings, DOI: 10.1002/tal.375.

[2] Popov, E., P. (1991)., ”Engineering Mechanics of Solids”,Prentice-Hall Inc.

[3] Chen, W., F., and Han, D., J. (1988)., ”Plasticity for Structural Engineers”, Springer-Verlag

New York Inc.

[4] DIANA Finite Element Analysis (2007)., Release 9.2, Users’s Manual, Material Library.,

DIANA TNO, BV.,

[5] Park, R. (1975)., ”Reinforced Concrete Structures”, John Wiley & Sons Inc.

[6] Rots, J.G., and Blaauwendraad, J. (1989)., ”Crack Models for Concrete: Discrete or

Smeared? Fixed Multi-Directional or Rotating?”, Heron, Vol. 34, p. 3-59

[7] Vecchio, F. J., and Collins, M. P. (1986)., ”The Modified Compression Field Theory for

Reinforced Concrete Elements Subjected to Shear.”, ACI Journal 83(22), p. 219-231.

[8] R. G. Selby, and F. J. Vecchio (1993)., ”Three-dimensional Constitutive Relations for

Reinforced Concrete Reinforced Concrete.”, Tech. Rep. 93-102, Univ. Toronto, Dept. Civil

Eng., Toronto, Canada

[9] Collins, M., and Porasz, A. (1989)., ”Shear Design for High Strength Concrete”,

ComiteEuro-International du Beton, Bulletin d’ Information No. 193, p. 77-83

[10] Gilbert, R., I. and Warner, R., F. (1978). ”Tension Stiffening in Reinforced Concrete

Slabs.” Journal of Structural Engineering, Vol. 104(12) p. 1885-1900

[11] Mander, J., B., Priestley, M., J., M. and Park, R. (1988)., ”Theoretical Stress-Strain Model

for Confined Concrete.”, Journal of Structural Engineering, Vol 114(8), p. 1804-1825

[12] Eligehausen, R., Popov, E., P. and Bertero, V., V. (1983)., ”Local Bond Stress-Slip Re-

lationships of Deformed Bars under Generalized Excitations.”, Report No. UCB/EERC-

83/23, University of Berkeley

BIBLIOGRAPHY 66

[13] FIB- Federation Internationale du Beton (2000)., ”Bond of Reinforcement in Concrete.”,

Chapter 1: ”Bond Mechanics Including Pull-out and Splitting Failure.”, (principal authors

Gambarova, P., G., Plizzari, G., A., Rosati, G., P. and Russo G.)

[14] Tepfers, R. (1979)., ”Cracking of concrete cover along anchored deformed reinforcing

bars.”, Magazine of Concrete Research, Vol.32(106), p. 3-12

[15] Nilson, A. H. (1971)., ”Bond Stress-Slip Relations in Reinforced Concrete.”, Report 345,

School of Civil and Environmental Engineering, Cornell University, Ithaca

[16] Fardis, M. W and Buyukozturk, O. (1980)., ”Shear Stiffness of Concrete by Finite Ele-

ments.”, ASCE Journal of the Structural Division, Vol. 101(ST6), p. 1311-1327

[17] Mirza, S., M. and Houde, J. (1978)., ”Study of Bond Stress-Slip Relationships in Rein-

forced Concrete.”, Journal of The American Concrete Institute, Vol 76(1), 978pp

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